In conclusion, we investigated the low temperature quantum behavior of SrTiO\(_3\)  from a fully microscopic point of view.
In line with the concept of quantum paraelectricity [1]}, we show that only the quantum description of the lattice predicts a stable phonon mode and hence the paraelectric phase; a result in contrast with conventional perturbation theory, which wrongly predicts an instability that leading to a ferroelectric transition at low temperature.
We show that, not only the quantum fluctuations of the FES phonon stabilize this phase, but that the interaction between FES mode and fluctuations of the \(c\)  lattice parameter contribute to the ground state and to the first FES eigenmode energy; therefore it is crucial to treat these degrees of freedom on the same footing in order to explain the dynamical properties of SrTiO\(_3\) .
Our DFT-based treatment of the lattice-nuclear Schrödinger equation provides low temperature phonon frequencies in agreement with the experimentally observed values.
Combined with thermal statistics, this treatment also reproduces the temperature dependence of the frequency of the FES mode and the dielectric constant, namely the frequency stiffening of the FES mode with increasing temperature and the flat behavior of the dielectric constant stemming from the quantum fluctuation of the lattice.
Besides providing a detailed and direct first principles description of the FES frequency and the quantum paraelectric ground state of SrTiO\(_3\) , which has been conjectured for a long time and thus far only been considered in the context of phase dynamics, this work opens new avenues to investigate recently observed light induced ferroelectricity in SrTiO\(_3\)  [2]}, [3]} and to address the ground state of SrTiO\(_3\)  embedded in an optical cavity [4]}, [5]} by providing a model for the low temperature nuclear lattice Hamiltonian.
